{
  "id": "1703.00901",
  "version": "v1",
  "published": "2017-03-02T18:59:02.000Z",
  "updated": "2017-03-02T18:59:02.000Z",
  "title": "A sharp Trudinger-Moser type inequality involving $L^{n}$ norm in the entire space $\\mathbb{R}^{n}$",
  "authors": [
    "Guozhen Lu",
    "Maochun Zhu"
  ],
  "comment": "33 pages, submitted for publication on February 8, 2017",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $W^{1,n} ( \\mathbb{R}^{n} $ be the standard Sobolev space and $\\left\\Vert \\cdot\\right\\Vert _{n}$ be the $L^{n}$ norm on $\\mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm \\[ \\underset{\\left\\Vert u\\right\\Vert _{W^{1,n}\\left(\\mathbb{R} ^{n}\\right) }=1}{\\sup}\\int_{ \\mathbb{R}^{n}}\\Phi\\left( \\alpha_{n}\\left\\vert u\\right\\vert ^{\\frac{n}{n-1}}\\left( 1+\\alpha\\left\\Vert u\\right\\Vert _{n}^{n}\\right) ^{\\frac{1}{n-1}}\\right) dx<+\\infty \\]in the entire space $\\mathbb{R}^n$ for any $0\\leq\\alpha<1$, where $\\Phi\\left( t\\right) =e^{t}-\\underset{j=0}{\\overset{n-2}{\\sum}}% \\frac{t^{j}}{j!}$, $\\alpha_{n}=n\\omega_{n-1}^{\\frac{1}{n-1}}$ and $\\omega_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $\\mathbb{R}^n$. We also show that the above supremum is infinity for all $\\alpha\\geq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $\\alpha>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work \\cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $\\Phi(t)$ is replaced by a strictly smaller $\\Phi^*(t)=e^{t}-\\underset{j=0}{\\overset{n-1}{\\sum}}% \\frac{t^{j}}{j!}$. (Note that $\\Phi(t)=\\Phi^*(t)+\\frac{t^{n-1}}{(n-1)!}$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-02T18:59:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp trudinger-moser type inequality",
      "entire space",
      "standard sobolev space",
      "dimensional surface measure",
      "nonlinear euler-lagrange equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}